Finite Group Databasebeta

← 3−1+2×#997 #72220095

Subgroups — 3−1+2×#997 #72220095

Order 2592.

610 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#997C311yesmaximal, characteristic
23−1+2×H(5,𝔽2)C311yesmaximal, characteristic, complemented
3#11722589C312yesmaximal, complemented
4C9⋊#997C316yesmaximal, complemented
5C9×#997C313yesmaximal, complemented
6C2×A4×3−1+2—43nomaximal
7SL(2,𝔽3)×3−1+2—42nomaximal
8#41×3−1+2—32no—
9C2×D4×3−1+2—33no—
10A4×3−1+2—43no—
11C32×H(5,𝔽2)C3211yescharacteristic
12C3×#997—33no—
13C3×#997C3212yes—
14C3×#997C3211yescharacteristic
15C9×H(5,𝔽2)C3213yescomplemented
16#63703C3216yescomplemented
17C2×C32×A4—43no—
18C32×SL(2,𝔽3)—42no—
19C23×3−1+2—31no—
20C23×3−1+2A413yescomplemented
21C2×#1094—46no—
22C2×C9⋊A4—418no—
23C2×C9×A4—49no—
24Q8×3−1+2A412yescomplemented
25D4×3−1+2—36no—
26C2×C4×3−1+2—33no—
27#6204—44no—
28C9⋊SL(2,𝔽3)—412no—
29C9×SL(2,𝔽3)—46no—
30C2×C3×3−1+2—161no—
31C32×#41—32no—
32C2×C32×D4—33no—
33C9×#41—36no—
34C2×D4×C9—39no—
35C32×A4—43no—
36C22×3−1+2—33no—
37C22×3−1+2—43no—
38C22×3−1+2—121no—
39#1094—46no—
40C9⋊A4—418no—
41C9×A4—49no—
42C4×3−1+2—32no—
43C3×H(5,𝔽2)—31no—
44C3×H(5,𝔽2)C3311yescharacteristic
45#997—32no—
46#9973−1+213yesdirect product factor
47C3×3−1+2—161noSylow
48C23×C32—31no—
49C23×C32C3×A413yes—
50C2×C3×A4—43no—
51C2×C3×A4—46no—
52C2×C3×A4—129no—
53C32×Q8C3×A412yes—
54C32×D4—36no—
55C2×C32×C4—33no—
56C3×SL(2,𝔽3)—42no—
57C3×SL(2,𝔽3)—44no—
58C3×SL(2,𝔽3)—126no—
59C23×C9—33no—
60C23×C9C3×A419yescomplemented
61C2×C22⋊C9—418no—
62Q8×C9C3×A416yescomplemented
63D4×C9—318no—
64C2×C4×C9—39no—
65Q8⋊C9—412no—
66C2×C33—161no—
67C2×3−1+2—63no—
68C2×3−1+2—162no—
69C2×3−1+2—166no—
70C2×3−1+2C22⋊A411yescharacteristic
71C2×C3×C9—163no—
72C3×#41—32no—
73C3×#41—92no—
74C2×C3×D4—33no—
75C2×C3×D4—93no—
76C22×C32—33no—
77C22×C32—43no—
78C22×C32—121no—
79C3×A4—43no—
80C3×A4—46no—
81C3×A4—129no—
82C32×C4—32no—
83C22×C9—39no—
84C22×C9—49no—
85C22×C9—123no—
86C22⋊C9—418no—
87C4×C9—36no—
88H(5,𝔽2)C3×3−1+211yescharacteristic, complemented, Sylow
89C33—161no—
903−1+2—162no—
913−1+2—166no—
923−1+2#99711yescharacteristic, direct product factor
93C3×C9—163no—
94C23×C3—31no—
95C23×C3—33no—
96C23×C3—91no—
97C23×C3C32×A413yes—
98C2×A4—49no—
99C2×A4—126no—
100C3×Q8—32no—
101C3×Q8C32×A412yes—
102C3×D4—36no—
103C3×D4—96no—
104C2×C3×C4—33no—
105C2×C3×C4—93no—
106SL(2,𝔽3)—46no—
107SL(2,𝔽3)—124no—
108C2×C32—63no—
109C2×C32—161no—
110C2×C32—162no—
111C2×C32—483no—
112C2×C32C3×C22⋊A411yescharacteristic
113C2×C9—69no—
114C2×C9—166no—
115C2×C9C3×C22⋊A413yes—
116#41—32no—
117C2×D4—33no—
118C22×C3—33no—
119C22×C3—43no—
120C22×C3—93no—
121C22×C3—123no—
122C22×C3—121no—
123C22×C3—361no—
124A4—49no—
125A4—126no—
126C3×C4—32no—
127C3×C4—92no—
128C32—161no—
129C32—162no—
130C32—483no—
131C32C3×#99711yescharacteristic
132C9—166no—
133C9C3×#99713yescomplemented
134C23—31no—
135C23A4×3−1+213yescomplemented
136Q8A4×3−1+212yescomplemented
137D4—36no—
138C2×C4—33no—
139C2×C3—31no—
140C2×C3—63no—
141C2×C3—163no—
142C2×C3—183no—
143C2×C3—482no—
144C2×C3C32×C22⋊A411yescharacteristic
145C22—33no—
146C22—43no—
147C22—121no—
148C4—32no—
149C3—31no—
150C3—163no—
151C3—482no—
152C3C32×#99711yescharacteristic
153C2—63no—
154C23−1+2×C22⋊A411yescharacteristic
subgroupG/H
center, Frattini, socleC2×C3C32×C22⋊A4
commutator, cosocleC3×H(5,𝔽2)C33
Fitting3−1+2×H(5,𝔽2)C3
prime radicalC32×#997C3
hypercenterC2×3−1+2C22⋊A4
hypocenterH(5,𝔽2)C3×3−1+2
LCSG ▷ C3×H(5,𝔽2) ▷ H(5,𝔽2)
UCSC1 ◁ C2×C3 ◁ C2×3−1+2

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. 3−1+2×H(5,𝔽2)C3semidirect
3. #11722589C3semidirect
4. C9⋊#997C3semidirect
5. C9×#997C3semidirect
15. C9×H(5,𝔽2)C32semidirect
16. #63703C32semidirect
20. C23×3−1+2A4semidirect
24. Q8×3−1+2A4semidirect
46. #9973−1+2direct (N × H)
60. C23×C9C3×A4semidirect
62. Q8×C9C3×A4semidirect
88. H(5,𝔽2)C3×3−1+2semidirect
92. 3−1+2#997direct (N × H)
133. C9C3×#997semidirect
135. C23A4×3−1+2semidirect
136. Q8A4×3−1+2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C32×#9973
3−1+2×H(5,𝔽2)3
#117225893
C9⋊#9973
C9×#9973
C2×A4×3−1+24
SL(2,𝔽3)×3−1+24